1. Field of the Invention
This invention solves a problem involving processing of seismic data that originate from transducers emplaced on an irregular surface. In a related problem, the transducers are positioned on a flat surface, but beneath the surface, there exists a velocity interface having substantial topographic relief and a concomitant significant velocity contrast.
2. Discussion of Related Art
Ordinarily, geophysical exploration of large areas is most readily conducted by emplacing the physical-measurement instrumentation on the surface of the earth. In some land areas, such as the Front Range of the Rocky Mountains, the topographic relief is a bit irregular. In other land areas, such as the Alberta syncline, the surface may be flat prairie but the underlying Paleozoic surface is severely eroded. Devonian reefs form a disconformity with respect to the underlying Silurian formation. At sea, the surface is usually flat but the sea floor may be topographically rugged with a substantial velocity contrast across the water/formation boundary.
Unless properly corrected, the raw geophysical measurements are distorted by the irregular topography of the earth at the surface or by raypath bending that occurs at the formation interfaces subsurface. That is, the quantitative geophysical measurements tend to display the approximate topography of the irregular interface instead of the desired topography of the underlying target formation.
By way of example but not by way of limitation, this disclosure will be narrated in terms of seismic exploration geophysics but the basic concepts apply equally well to other geophysical studies such as potential field or electromagnetic measurements.
As is well known in seismic surveying a plurality of transducers are distributed over the surface of an area under study. A separate source or a certain transducer is selected as a source for generating an acoustic wavefield while the remaining transducers serve as receivers of the acoustic waves after reflection from subsurface formations. The traveltime from the source to each of the respective receivers, corrected for non-vertical raypaths and multiplied by half the average earth velocity, is a measure of the distance between the receiver and the reflecting interface. If the reflecting interface is flat and the surface is irregular, the travel time will vary in proportion to the surface elevation, producing an incorrect image.
Traditionally, to correct for surface topography, a horizontal datum is established, usually at an elevation above the tallest topographic feature. The seismic travel times are extrapolated upwards from the surface to the datum by applying a static time correction equal to the vertical distance between surface and datum divided by one-half of a preselected replacement velocity. The replacement velocity may be selected as an extension of the near-surface velocity function characteristic of the region under study. An analogous process may be implemented to correct for subsurface topography. Reduction of the seismic reflection times to a reference plane may be referred to variously as datumizing or datuming.
When a large interval exists between the surface and the datum, a strictly vertical static correction path does not fit the actual ray path geometry that would have existed if the recording surface had coincided with the datum. That discrepancy is shown in FIG. 1. In FIGS. 1 and 2, the ordinate is depth and the abscissa is transducer offset. Using the well-known exploding reflector concept as illustrative for a point diffractor, the raypaths such as 10 emerging at the recording surface 12 should be extrapolated upwards to the datum 13 along the dashed slant paths such as 14 of FIG. 1 rather than along the dashed vertical paths such as 16 as shown in FIG. 2. A pure vertical time shift as in FIG. 2 simply displaces the diffraction hyperbola 18 (FIG. 3B) bodily to a position deeper in time to 20 but lacking a corresponding flattening in the slopes (that is, a reduction in the normal moveout) of the limbs 22 and 24 as would be expected had the diffractor in fact existed at a greater depth. In FIGS. 3A and 3B, the ordinate is wavefield traveltime and the abscissa is lateral transducer location. Limb flattening is illustrated in FIG. 3A which shows hyperbola 18 from FIG. 3B and hyperbola 21 from a deeper bed with the expected reduction in normal moveout at limbs 23 and 25 due to the greater raypath lengths. Application of a wave-equation migration algorithm, applied from the datum, to the constant-time-displaced diffraction hyperbola 20 of FIG. 3B would result in over-migration.
Berryhill, in 1979, discussed the problem of wave-equation datuming in a paper entitled "Wave-equation datuming" published in Geophysics, v. 44, pp. 1329-1344 with a follow-up Note in 1984 entitled "Wave-equation datuming before stack" Geophysics, v 49, pp. 2064-2066. He provides an exact solution to the datuming problem but the main difficulty with Berryhill's method is that it is so computationally intensive as to be uneconomical.
In U.S. Pat. No. 4,943,950, issued Jul. 24, 1990, and assigned to the assignee of this invention, Beasley et al., teach a more computationally efficient method for datumizing seismic data. The '950 patent teaches a method for migrating seismic data that has been vertically static corrected. For data below a datum but above the surface elevation, a zero (or very small) migration velocity is applied. A best estimated earth velocity is used to migrate the data below the original data-recording surface.
For good and sufficient reasons, use of the Beasley et al. method is confined to the use of a finite difference migration algorithm. That algorithm is a much less accurate and less efficient method than cascaded f-k migration or any algorithm that operates in the wavenumber domain such as phase-shift migration. However, algorithms that operate in the wavenumber domain are not amenable to migration from the surface using the initial zero-velocity layer of the Beasley et al. method. That is why the finite difference algorithm is required.
There is a need for an economical and efficient datumizing method that will allow use of any desired migration algorithm for subsequent processing.